Stochastic Models for Fractional Calculus

By Mark M. Meerschaert, Alla Sikorskii. 2011

765 citations recorded by the source corpus.

Preface: 1 Introduction 1.1 The traditional diffusion model 1.2 Fractional diffusion 2 Fractional Derivatives 2.1 The Grunwald formula 2.2 More fractional derivatives 2.3 The Caputo derivative 2.4 Time-fractional diffusion 3 Stable Limit Distributions 3.1 Infinitely divisible laws 3.2 Stable characteristic functions 3.3 Semigroups 3.4 Poisson approximation 3.5 Shifted Poisson approximation 3.6 Triangular arrays 3.7 One-sided stable limits 3.8 Two-sided stable limits 4 Continuous Time Random Walks 4.1 Regular variation 4.2 Stable Central Limit Theorem 4.3 Continuous time random walks 4.4 Convergence in Skorokhod space 4.5 CTRW governing equations 5 Computations in R 5.1 R codes for fractional diffusion 5.2 Sample path simulations 6 Vector Fractional Diffusion 6.1 Vector random walks 6.2 Vector random walks with heavy tails 6.3 Triangular arrays of random vectors 6.4 Stable random vectors 6.5 Vector fractional diffusion equation 6.6 Operator stable laws 6.7 Operator regular variation 6.8 Generalized domains of attraction 7 Applications and Extensions 7.1 LePage Series Representation 7.2 Tempered stable laws 7.3 Tempered fractional derivatives 7.4 Pearson Diffusion 7.5 Classification of Pearson diffusions 7.6 Spectral representations of the solutions of Kolmogorov equations 7.7 Fractional Brownian motion 7.8 Fractional random fields 7.9 Applications of fractional diffusion 7.10 Applications of.

Subfields: Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/ba32a55de8a942b6d2dfa39892f4d87a52836705

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Citations in the corpus

A characteristic difference method for the variable-order fractional advection-diffusion equation (2013, neutral): “[18]) and hydrology (Liu et al.”
Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups (2013, background): “When the function f is f(λ) = λ, α ∈ (0, 1), the related subordinator is called the α-stable subordinator and the inverse process L(t) = inf {s > 0 : σ(s) > t} is called the inverse stable subordinator (see Meerschaert and Sikorskii [24]; Meerschaert and Straka [25]; Samorodnitsky and Taqqu [33] for”
What is the fractional Laplacian? A comparative review with new results (2018, methodology): “Overview During the past few decades, scientists have been exploring fractional calculus as a tool for developing more sophisticated mathematical models that can accurately describe complex anomalous systems [1, 2, 3, 4].”
Time fractional equations and probabilistic representation (2017, background): “1 Introduction Fractional calculus has attracted lots of attentions in several fields including mathematics, physics, chemistry, engineering, hydrology and even finance and social sciences (see [9, 20, 22, 21]).”
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients (2015, background): “Introduction Fractional calculus has been used in numerous areas including mathematical modeling [24, 44], control engineering [6, 32], electromagnetism [13, 43], polymer science [3, 29], hydrology [4, 41], biophysics [14, 22], and even finance [36, 40].”
Machine Learning of Space-Fractional Differential Equations (2018, background): “Moreover, unlike standard derivatives, fractional derivatives, whether in R, R, or on bounded subsets, are nonlocal operators typically defined by singular integrals or eigenfunction expansions that are difficult and expensive to discretize [14], [18].”
Fractional Calculus: Quo Vadimus? (Where are we Going?) (2015, result): “We have taken the present occasion, as a compliment to our previous survey [107], to cite books published during the years 2011 [92, 80, 44, 26, 65, 47, 5, 106, 117], 2012 [7, 3, 77, 1, 51, 38, 102, 56, 122, 15, 112, 110, 118], 2013 [14, 18, 89, 29, 36], and 2014 [57, 4, 121, 39, 105, 40, 66, 86, 2,”
The fractional non-homogeneous Poisson process (2016, methodology): “…probabilities where the three-parameter generalized Mittag-Leﬄer function is deﬁned as follows We will use the fractional Caputo-Djrbashian derivative which is deﬁned as (2.13) Its Laplace transform is Equations (2.13), (2.14) and (2.15) can be found in Meerschaert and Sikorskii (2012, p. 34).”
Well-posedness and numerical algorithm for the tempered fractional differential equations (2019, background): “For the other different definitions of the tempered fractional integration, see the books [43, 38, 31] and references therein.”
Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems (2013, background): “tings used in our tests. Case 1 We consider Ω = (−1,1), ΩI = (−1−ε,−1)∪ (1,1+ε), γ1(x,y) = 1 |x−y|1+2s , s∈ (0,1), (52) 5Numericaltests 14 for which L corresponds to a fractional diﬀerential operator [40], and the data set A : f(x) = 1 bu(x) = uA(x) u(x)|Ω I = 0, s= 0.7. (53) Here uA(x) is a su”
Fractional Skellam processes with applications to finance (2014, background): “see, for example, Meerschaert and Sikorskii (2012). Note that E (t), t ≥ 0 is non-Markovian with non-stationary and non-independent increments.”